I have a very short and probably very stupid question. While doing an exercise I bumped into the following relation:
If two vectors $\vec{x}$ and $\vec{y}$ are parallel, then $\vec{x} =$ $ <\vec{x},\vec{y}>\vec{y}$.
I don't know if this relation always holds, but if it does, can someone maybe verify for me how? It looks super easy but I can't seem to figure it out.
Thanks in advance!
On
The projection of vector $\vec x$ onto vector $\vec y$ is $\dfrac {<\vec x,\vec y>}{<\vec y,\vec y>}\vec y$. When $\vec x$ and $\vec y$ are parallel, this is $\vec x$. If $\vec y$ is a unit vector, then $<\vec y,\vec y>=1$ and the relation you gave holds.
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If $\vec y=c\vec x$, then the projection is $\dfrac{<\vec x,c \vec x>}{<c\vec x,c\vec x >}c\vec x=\vec x$
The relation $$\vec{x} =<\vec{x},\vec{y}>\vec{y}$$ is not true in general even if the vectors are parallel because the norms do not match.
However we can fix it by slightly changing it to $$\vec{x} =\frac {<\vec{x},\vec{y}>}{||y||^2}\vec{y}$$ which is true if the vectors are parallel.
Note that the vectors the same directions and and in case of being parallel they also have the same norm.